A. Field of the Invention
The present disclosure generally relates to fractional order capacitors having improved control over capacitor characteristics, including the complex impedance phase angle. The capacitors have a dielectric nanocomposite layer with filler material which allows for improved control over the capacitor characteristics.
B. Description of Related Art
Historically, fractional order calculus has been unexplored in engineering, because of its complexity and the fact that it did not have a fully acceptable geometrical or physical realization. For example, electrical components are generally limited to the specific characteristics of ideal inductors, resistors, and capacitors which have α values of −1, 0, and 1 respectively. In this context, α may be used to determine the phase shift between a device's current and its voltage through the equation −απ/2. When converted to degrees, i.e. 90, 0 and −90, these values represent the complex impedance phase angle.
However, since electrical components do not have exact integer values of a when implemented in real applications, having the ability to intentionally control the value of αsomewhere between the characteristics of the standard components may be useful for a variety of applications including performing automatic control, pattern recognition, system characterization, signal processing, and applying filters and oscillators related to the fields of electrochemistry, viscoelasticity, and even biological and neural systems.
Previous attempts to intentionally design electrical components with specific α values within the range 0<α<1, have generally resulted in very bulky designs that in many cases are not usable or practical in real applications or circuits. For example, with respect to liquid-electrode-based (LEB) type fractional capacitors, copper electrodes are immersed in a PMMA-choloroform solution, and the phase angle is varied depending on the depth of immersion of the electrodes. However, this method clearly does not allow integration with printed circuit boards and/or electronic circuits easily. Moreover, packaging of such a setup results in a very bulky apparatus.
Other attempts have included fractal-type (FT) fractional capacitors designs. These designs are typically created on wafers and rely on transmission line theory. The basic operating principle behind these types of capacitors involves creating fractal geometries as stubs, or transmission lines, which in turn can yield a specific impedance based on the geometry and the technological parameters. Instead of creating a capacitor, the FT uses a series of metal traces that are created on the circuit to create the impedance. This is often referred to as a distributed element design as opposed to a lumped element design. Moreover, when using FT fractional capacitor, the values of α that can be achieved are only in the range of 0.46-0.5. Further, the constant phase behavior occurs at very high frequency ranges (1 MHz-10 GHz) given the nature of microwave circuits, and the variation in the phase angle is around 5°.
Other approaches have included simulating by digitally approximating the fractional order problems and calculating approximate solutions. Digital approximations are necessarily limited in bandwidth, highly consumptive of computer resources, and can suffer from numerical instabilities due to finite precision arithmetic. These limitations can make digital techniques impractical or incapable of solving many problems, such as controlling fast processes or “stiff” processes, which involve strong opposing forces.